Convergence rate improvement for adaptive receiving array antennas of higher order than 2-pulse mti cancellers

ABSTRACT

A system for improving the convergence rate of adaptive receiving array antenna systems in MTI radar systems, having cancellers of higher orders than a 2-pulse canceller, is provided by a transformation matrix according to the following technique. Form the matrix H with elements

United States Patent 1 Brennan et al.

[ 51 Apr. 3, 1973 [541 CONVERGENCE RATE IMPROVEMENT FOR ADAPTIVERECEIVING ARRAY ANTENNAS OF HIGHER ORDER THAN Z-PULSE MTI CANCELLERS[75] Inventors: Lawrence E. Brennan, Tarzana; Irving S. Reed, SantaMonica, both of Calif.

[73] Assignee: Technology Service Corporation,

Santa Monica, Calif.

[22] Filed: Sept. 13, 1971 [21] Appl. No.: 179,782

[52] US. Cl. ..343/7 A, 343/7.7 [51] Int. Cl .L ..G01s 9/42 [58] Fieldof Search ..343/7 A, 7.7

[56] References Cited UNITED STATES PATENTS 3,417,396 12/1968 Stifter etal ..343/7.7 3,587,097 6/1971 Stull ..343/7 A Primary ExaminerT. H.Tubbesing AuomeyLindenberg, Frilich & Wasserman ABSTRACT pulsecanceller, is provided by a transformation matrix according to thefollowing technique.

' Form the matrix H with elements where e log l/p and p is thecorrelation between successive returns. The transformation matrix thenhas the form where GFAC is a gain factor greater than unity, i]; is aphase shift angle, and h,, h, h are the eigenvectors of H ordered suchthat the associated eigenvalues are monotically increasing. Antennaelement signals V are then transformed according to the equation X UV.

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SHEET 1 BF 5 INVENTORS.

LAWRENCE E. BRENNAN BY IRVING S. REED pzfnw W/W ATTORNEY PAIENTEDr-m ma3.725822 sum 3 u? 5 MTI GAIN (DB) MTl 9 I I I I o 4 8 4 l2 l6 2o SAMPLES(x10 FIG. 3

GAIN (DB) PATENTEDAEM- m5 SHEET 5 BF 5 MTl GAIN CONVERGENCE RATEIMPROVEMENT FOR ADAPTIVE RECEIVING ARRAY ANTENNAS OF HIGHER ORDER THANZ-PULSE MTI CANCELLERS BACKGROUND OF THE INVENTION This inventionrelates to a method and apparatus for adaptive receiving array antennasfor moving-target-indicating (MTI) systems, such as in airborne radar orsonar systems, and more particularly to a method and apparatus forimproving the convergence rate of adaptive receiving array antennasystems having cancellers of higher orders than a two-pulse canceller.

In a copending application Ser. No. 179,777 filed on Sept. 13, 1971 bythe same inventors and titled ADAP- TIVE RECEIVING ARRAY ANTENNA METHODSAND APPARATUS FOR MTI SYSTEMS, a method of adaptive signal processing inairborne search radar and the like is described. That method employsadaptive control loops for producing complex weights W where n denotesthe n" antenna element channel from 1 to N and k denotes the k" pulse ofa set of K consecutive samples obtained from each of the N arrayelements. These KN coherent samples (both amplitude and phaseinformation is retained) of the radar return, Vnle, are multiplied bythe adaptively controlled complex weights and added to obtain the output1': for the corresponding range resolution cell.

The set of complex weights {W are produced by adaptive control loopswhich are identical except for differences in the steering signals.Steering signals for a K pulse system, matched to a target dopplerfrequency f are n=l,2 N S -znnkr k=l,2 k (1) when the main beam scanangle is normal to the linear array antenna. At other scan angles aphase gradient is introduced along the array appropriate to the desiredscan angle. In a two-pulse (K=2) MTI radar system with the line of sight(antenna axis) normal to the platform velocity, the set {W,,,,} isformed adaptively with the steering signals 8", =1, 8* l (n=l, 2 N). Thecorresponding steering signals for a three-pulse (K 3) MTI system areS*,,' l, S",, l and S*,, 1. At other angles, the S*,,,, are matched to atarget moving M4 radially between pulses.

It was found that excellent MTI gain can be achieved for all scanangles. However, in many cases of interest, the convergence rate of theadaptive system is too slow. A highly effective means for speedingconvergence in a two-pulse system is described in the aforesaidcopending application. It would be desirable to provide an equallyeffective means for speeding convergence in a higher order pulse system.

Briefly, the invention of the aforesaid copending application isembodied in a coherent pulsed airborneradar system. A separate coherentoutput is available from each of N elements of a linear array, where anelement may be a single slot or dipole, or plurality of slots or dipolescombined in a conventional manner. A set of NK coherent returns orsamples are multiplied by adaptively controlled complex weights andadded to obtain the output for the corresponding range resolution cell.To accomplish that, each of the NK samples is coupled to a summing meansby an adaptive loop. The output of the summing means is given by whereW,,,, is the weight applied to the n'" element for the k pulse of thetrain, and V is the corresponding sample of the received voltage. Thesteady state weights are given by where W denotes a column vector of theweights W M is the clutter covariance matrix, I the identity matrix, Gthe loop gain, and S* the steering vectors.

To improve the convergence rate of a two-pulse MTI system, the twooutputs V, and V from each antenna array element are transformed asfollows:

Where rl1=-41rD cos 0 A, D is the distance the platform moves betweenpulses, 0 is the main beam scan angle from the radar velocity vector andA is the wavelength.

SUMMARY OF THE INVENTION It has been discovered that the transformationmatrix of the aforesaid copending application can be expanded from atwo-pulse MTI system to a higher order K-pulse system, i.e., where k isan integer greater than 2 by forming the matrix H with elements where elog Up and p is the correlation between successive returns. Let h,, h hh be the eigenvectors of H, ordered such that the associated eigenvaluesare monotonically increasing. The order of the matrix is equal to thenumber K of pulses in the higher order system. The transformation matrixU has the form where GFAC is a gain factor greater than unity and i1: isa phase shift angle. Equation (8) indicates that the h are rows of U Theantenna element signals V are then transformed according to the equationX UV (9) The phase shift factor e in the matrix U is for scan anglesother than normal to the antenna velocity vector. Different gains areemployed in the different X- signal channels, either in the last stageof the transformation matrix, at the input terminals of followingadaptive control loops, or in the adaptive control loops (where the gainG is then made equal to GFAC The phase shift angle is a function ofantenna velocity and scan angle, and is selected to cancel clutter atthe center of the beam. In Equation (10), D is the distance the antennamoves between pulses, is the scan angle relative to the antenna velocityvector, and I. is wavelength.

BRIEF DESCRIPTION OF THE DRAWINGS FIG. 1 is a block diagram of thepresent invention embodied in a three-pulse MTI radar system.

FIG. 2 is a block diagram of an adaptive control loop for use in FIG. 1.

FIGS. 3 to 5 are plots of MTI gain vs. number of samples for variouscases of MTI three-pulse canceller simulation with the presentinvention.

DESCRIPTION OF THE PREFERRED EMBODIMENTS Referring now to FIG. 1, atransformation matrix for improving the convergence rate of athree-pulse airborne MTI radar system is implemented according to thefollowing equations:

where, as noted hereinbefore, 41 41rD cos 6M and GFAC l. It should benoted that Equations (11) to (13) are but an application of Equations(7) to (9) to the special case of a three-pulse canceller where theorder of the matrix U is equal to 3, and that U is unitary for GFAC 1,Le. UU* I, the identity matrix.

The radar receiving system includes an array of N elements disposed in aline at any scan angle. Each element may be a single slot in a waveguideor a dipole, but in the normal system will be a linear array of slots ordipoles disposed in a line normal to the array of N elements andcombined in a standard manner to provide a single output V, where ndenotes a given one of the elements I to N.

Each output signal V, is transmitted through two delay elements toproduce three pulses V V,, and V,, in a conventional manner fordouble-delay (3 pulse) MTI cancellers. In the example of FIG. 1, element1 produces the signal V which is amplified by a low noise amplifier 11and transmitted through delay elements 12 and 13 to produce signals V Vand V These signals are then transformed by a matrix 10 into the signalsX X and X according to the foregoing Equations (11), (12) and (13)which, as just noted, are the special case for a three-pulse MTIcanceller system derived from the general equations (7), (8) and (9).The transformation matrix 10 thus transmits the signals X X and X withthe following values For scan angles normal to the radar platformvelocity direction, the phase shift angle .1. is zero. Accordingly, thetransformed signals are simply FAC/ 6) 2 V...+ V...)

trol loops are also identical except for steering signals That outputsignal is fed back to all of the adaptive control loops to form the setof complex weights (W For convenience in understanding the presentinvention, the adaptive control loop for a given complex weight Wdescribed in the aforementioned application is shown in FIG. 2. Thecontrol loop is comprised of a multiplier 21 and low pass filter 22which provide a cross-correlation u between X*,,,, and the output V; Inchannels where this correlation is large, the complex weight at theoutput of an amplifier 23 will change more rapidly than in channels oflow correlation. The complex weight is formed by the amplifier 23according to the following equation:

where G is a gain factor of the amplifier. The conjugate X*,,,. is usedto form the cross-correlation and is obtained from the complex conjugateconverter 25. V x is derived from the signals X each multiplied by anassociated weight W Accordingly, each signal X is applied to amultiplier 24 to form the product W X The sum of the products is theoutputs V:

The transformation matrices amplify the X components by a gain factorGFAC relative to the other components X,, and X,,;,. That may beaccomplished in the output stages of the transformation matrices, or inthe adaptive control loops for the X,, components by making the gainfactor G (of amplifier 23) equal to the square of the gain factor GFAC.

The signal return from a target moving at optimum radial velocity, i.e.moving at M4 radial motion between pulses, has the following form in Vspace:

SOC -c' 2O Accordingly, the steering signals in V space are proportionalto S*, and since a signal return could be transformed into the X-spacesystem by a matrix U, the optimum steering signals in X space areproportional to the complex conjugate of this X-space signal vectoraccording to the following.

The transformation matrix U appropriate for a threepulse system, wasfound by computing the eigenvectors of the matrix S*oc Anytransformation matrix could be used in place of the matrix U; theobjective is to find a transformation matrix which improves performancewhen different gains are used in the transformed channels. Anynonsingular transformation matrix preserves all of the input informationand admits any solution in transformed space which could be achieved inthe original V space. The transformation matrix of Equation (12) is agood choice, since the X components contain minimum clutter and thetransformation is unitary.

Toreiterate, in implementing this improvement of convergence rate forthe special case of a three-pulse canceller, the steps are:

l. Transform the three consecutive outputs from each array element andeach range cell (V V V to (X,,,, X X using the transformation U ofEquation (l2). The transformation includes in the formation of the X,components a gain factor GFAC 1 relative to any gain factor in theformation of the X, and X components.

. Apply the quantities X X X n=1 ,2 N, to a set of 3N adaptive loopswith the steering signals of Equation 24).

The output V is the weighted sum of the individual control loop inputs,as follows:

The space-time adaptive system according to FIG. 1 with N=l0 wassimulated. Approximately 30 cases were run to determine optimumparameters and to estimate the convergence rate improvement. Table Ibelow lists some of the results obtained after 500 range cells ofclutter and 500 adaptive loop iterations were simulated. The casesmarked with an asterisk are plotted in FIGS. 3, 4'and 5, and the curvesare identified by number corresponding with the lines in the table. Itshould be noted that the gain scale is different for each curve. Thefirst two cases (FIG. 3) illustrates the improvement in convergence ratefor an 0 scan angle, i.e., main beam oriented along the radar platformvelocity vector. After 2000 samples of clutter were processed, the MTIgain increased from roughly 48 dB to dB. Note that the loop timeconstant 1' was increased (inall loops) by a factor of 3 when GFAC was10. This increase in 'r was required to maintain loop stability.

TABLE I [Simulation results for 3-pulse array Doppler processing withconvergence rate fix] 500 sample 500 sample Loop Steady Sean MTI gain-MTI gainnoise State angle, simulated calculated, factor, gain, degreesGFAC decibels decibels decibels decibels Comments 0 0X10 I 45. 7 47. 2l. 42 79. 1 '0 27x10 10 5g. 1 58. J l. 38 101 Q) 4 .J l 45 9x10 1 l 55 l0 42 2 l2( r)uns with different random input.

. J 45 27x10 10 i 56. 7 2 48 0 {Verified without transformation. 459X10" 3 48. 3 53. 1 1. 50 00. 0 45 9X10 10 Unstable 59. 7 2. 46 96. 0 4581 10 30 Unstable 58. 8 2.09 97. 3 45 162x10 30 40. 5 48. 7 1. 54 97. 345 9X10 1 43. 0 44. 1 l. 28 81. 4 2 pulse. 45 27X10 10 47. 3 51. 5 1. 6287. 3 Do. 45 9X10 10 Unstable 53. 2 2. 86 87.3 Do. 90 9X10 1 51. 2 1. 4287. 2 90 27 10 10 Unstable 59. 7 1. 93 93. 7 90 54 (10 10 55. 58. 7 1.47 93. 7

'(usvs shown in' plots.

8 ulnnnnts. Loop gnin=l00. 0.3.\ motion between pulses.

FIG. 4 shows the respective simulation results obtained with and withoutthe transformation for a 45 scan angle. The MTI gain after 2000 loopiterations is improved from roughly 49 dB to 57 dB by the transformationmatrix. The improvement after 500 iterations is approximately 10 dB.This SOO-iteration figure is significant in these examples since theparameters GFAC and 1' were selected emperically based on SOO-samplesimulation runs.

The improvement for the cases of Nos. 12 and 14 with a 90 scan angle isillustrated in FIG. 5. The MTI gain increases from roughly 54 dB to 58dB after 2000 iterations due to the convergence rate improvementproduced by the transformation matrices. The improvement after 500iterations (again, the point for which parameters were selected) is 8dB.

In each of the cases simulated, the antenna platform motion betweenpulses was 0.3x. For the assumed eight-element array, this correspondsto about 0.075 antenna lengths of motion between pulses. This is alarger interpulse motion (higher radar platform velocity in terms ofradar wavelength) than in the simulation of a two-pulse system with thetransformation matrices of the aforesaid application. Three examples oftwo-pulse system performance are shown in cases 9,

10 and 11 of Table I for purposes of comparison. Note that 4 dB ofimprovement was obtained with the twopulse system at 45 scan angle. Forthe same parameters, the three-pulse system provides 10 dB of MTI gainimprovement after 500 loop iterations.

It appears that the effectiveness of this type of transformation matrixto improve convergence rate depends on the amount of clutter in the Xoutputs. The twopulse system is less effective in reducing the X,clutter than the three-pulse system. This suggests that, for smallinterpulse motions, the two-pulse system of the aforesaid applicationwill provide good transient response. As the interpulse motionincreases, three consecutive pulses must be processed in transformedspace with different gains to achieve good transient response. At evenhigher interpulse motions, a larger number of pulses may be required tomake this particular type of transformation useful.

The simulation was run without the transformation to X space for onecase and gave results identical to the GFAC l case in X space. Thiscorrespondence would be expected and serves as a check on the simulationprogram.

The loop noise factor is shown for each of the cases listed in Table I.This is the factor by which the loop noise increases the total outputnoise power under steady-state conditions. The calculated transientresponse and steady-state MTI gain are also shown for each case in TableI. The steady-state MTI gain improves for GFAC l since the bias error isreduced. This bias error results from the U6 term in W (M+I/G"S*, forthe steady state weights where W denotes a column vector of the weightsW,,,,, M is the clutter covariance matrix, I the identity matrix, G thecontrol loop gain, and S* the steering vectors.

The eigenvalue of Equation associated with the first row of Equation 12is very small as e 0; that associated with the second row isintermediate; and the eigenvalueof the third eigenvector (row) is large.This suggests using a gain greater than unity in both the X, and Xchannels, with larger gain in X channels. This was tried in two runswith no improvement over the cases shown in Table I. In one case, the X,and X channel gains were adjusted so that all diagonal elements of theresulting 3N X 3N covariance matrix were approximately the same. A gainof 6 was used in the X, channels and of 3.4 in the X channels toequalize all diagonal elements. At 45 scan angle and a loop noise factorof 1.57, the SOD-sample MTI gain was only 51.6

dB. This is not as good as the result shown for the same case in FIG. 4.

From the foregoing, it is evident that the convergence rate of an MTIcanceller can be improved by a transformation matrix of a higher orderthan a twopulse canceller. The example of a three-pulse (doubledelay)canceller given by Equations (II) to (13) or (14) to (16) serves toillustrate the method and apparatus. The apparatus would, of courseconsist of means for phase shifting the signals V,, and V,,,, by anglesill and 21:, respectively, and a plurality of adders, subtractors andmultipliers for carrying out the arithmetic operations indicated bythose equations. The gain factor GFAC may be introduced in the outputstage of the transformation matrix, or as a gain G in the adaptivecontrol loops, where G GFAC For higher order cancellers, similartransformation matrices produced according to the Equations (7) to (9)are used. Digital, analog or hybrid (digital-analog) techniques may beemployed in the implementation of the matrices for any order.

What is claimed is:

1. A method for improving the convergence rate of adaptive receivingantenna systems for moving target indication from signal return in whichNK samples of signal return are received, retaining both amplitude andphase information in each sample, where N is the number of receivingelements in an antenna array from one to any integer greater than one,and K is a number of consecutive samples from each array element andeach range cell from three to any integer greater than three for a pulsecancellation of higher order than two pulses, and in which each of saidNK samples is multiplied by adaptively controlled complex weights Wbefore summing samples to obtain an output according to the equation I(IF/H. l) I) l I) I) I) l) l l) l l) I) l) I) IN I) l) (l W l) l) I) l)I) l) I) gum m0 where GFAC l p is a phase shift angle, and h h, h arethe eigenvectors of H ordered such that the associated eigenvalues aremonotically increasing, and H is a matrix formed with elements H,,,,,=le|mn| where s log l/p and p is the correlation between successivereturns, thus indicating that the h,, are rows of U and where the phaseshift factor e"' in the matrix U is for scan angles other than normal tothe antenna velocity vector, and the phase shift angle i1; is equal to41rD cos /)t where D is the distance the antenna moves between pulsestransmitted for the antenna to receive as return signals, 0 is theantenna scan angle relative to the antenna velocity vector, and A iswavelength of the transmitted signal, and forming said complex weightsW,,,, from said output V; obtained by summing products W X andtransformed signals 2. A method for improving the convergence rate ofadaptive receiving antenna systems for moving target indication fromreturn signals according to claim 1 using a double-delay cancellerwherein K is equal to three, whereby U is the transformation matrixhaving the form nl nl X X z V V 2 3. A method as defined in claim 2wherein said scan angle 0 is 90 relative to the direction of motion,whereby transformed signals X,,,, X and X are computed according to thefollowing equations:

4. Apparatus for improving the convergence rate of an adaptive receivingantenna system for moving-target indication from a transmitted signalreturn, in which system NK samples V of signal return are received,

where N is the number of receiving elements in an antenna array from oneto any integer greater than one,

and K is a number from three to any integer greater than three ofconsecutive samples from each array element and each range cell, and inwhich each of said samples V is multiplied by adaptively controlledcomplex weights W before summing to obtain an output according to theequation where W,,,, is said complex weight applied as a multiplier tothe :k'" return of the n" element com rised of means for transforming Kconsecutive outputs V,,,, from each array element and each range cellaccording to the equation where U is a transformation matrix having theform where GFAC 1, this a phase shift angle, and h,, h h are theeigenvectors of H ordered such that the associated eigenvalues aremonotonically increasing, and H is a matrix formed with elements.

where e log Up and p is the correlation between successive retu-ms, thusindicating that the h, are rows of U and where the phase shift factor (ein the matrix U is for scan angles other than normal to the antennavelocity vector, and the phase shift angle \11 is equal to 41rD cos 0A,where D is the distance the antenna moves between pulses transmitted forthe antenna toreceive as return signals, 0 is the antenna scan anglerelative to the antenna velocity vector, and A is wavelength of thetransmitted signal, and forming said complex weights W from said outputV; obtained by summing products W,,,, X "k and transformed signals meansfor forming said complex weights W from outputs X fromsaidtransformation matrix and said output V means for multiplying outputsX,, from said transformation means-by respective weights W to formproducts W X and means for summing said products to form said output Vaccording to the equation 5. Apparatus as defined in claim 4 wherein Kis equal to three for a double-delay canceller, whereby U is thetransformation matrix having the form 6 Apparatus as defined in claimwherein said scan angle 6 is 90 relative to direction of motion wherebytransformed signals X X and X,, are computed according to the followingequations: 5

1. A method for improving the convergence rate of adaptive receivingantenna systems for moving target indication from signal return in whichNK samples of signal return are received, retaining both amplitude andphase information in each sample, where N is the number of receivingelements in an antenna array from one to any integer greater than one,and K is a number of consecutive samples from each array element andeach range cell from three to any integer greater than three for a pulsecancellation of higher order than two pulses, and in which each of saidNK samples is multiplied by adaptively controlled complex weights Wnkbefore summing samples to obtain an output according to the equation 2.A method for improving the convergence rate of adaptive receivingantenna systems for moving target indication from return signalsaccording to claim 1 using a double-delay canceller wherein K is equalto three, whereby U is the transformation matrix having the form
 3. Amethod as defined in claim 2 wherein said scan angle theta is 90*relative to the direction of motion, whereby transformed signals Xn1,Xn2 and Xn3 are computed according to the following equations: Xn1(GFAC/ Square Root 6) (Vn1 - 2Vn2 + Vn3) Xn2 (1/ Square Root 2) (Vn1 -Vn3) Xn3 (1/ Square Root 3) (Vn1 + Vn3)
 4. Apparatus for improving theconvergence rate of an adaptive receiving antenna system formoving-target indication from a transmitted signal return, in whichsystem NK samples Vnk of signal return are received, where N is thenumber of receiving elements in an antenna array from one to any integergreater than one, and K is a number from three to any integer greaterthan three of consecutive samples from each array element and each rangecell, and in which each of said samples Vnk is multiplied by adaptivelycontrolled complex weights Wnk before summing to obtain an outputaccording to the equation
 5. Apparatus as defined in claim 4 wherein Kis equal to three for a double-delay canceller, whereby U is thetransformation matrix having the form
 6. Apparatus as defined in claim 5wherein said scan angle theta is 90* relative to direction of motionwhereby transformed signals Xn1, Xn2 and Xn3 are computed according tothe following equations: Xn1 (GFAC/ Square Root 6) (Vn1 -2Vn2 + Vn3) Xn2(1/ Square Root 2) (Vn1 - Vn3) Xn3 (1/ Square Root 3) (Vn1 + Vn2 + Vn3)